An Information Theoretic Approach to Econometrics Paperback

Cover......Page 1
AN INFORMATION THEORETIC APPROACH TO ECONOMETRICS......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 17
1.1 Book Objectives and Problem Format......Page 19
1.2 Organization of the Book......Page 21
1.3 Selected References......Page 22
PART I TRADITIONAL PARAMETRIC AND SEMIPARAMETRIC ECONOMETRIC MODELS: ESTIMATION AND INFERENCE......Page 23
2.1 Data Sampling Processes (DSPs)\nand Notation......Page 25
2.2 A Parametric General Linear Model......Page 28
2.2.1 The Parametric Model and Maximum Likelihood (ML) Estimation of β and σ2......Page 30
2.2.2 The Parametric Model and Inference......Page 33
2.3 A Semiparametric General Linear Model......Page 34
2.3.1 The Squared Error Metric and the Least Squares (LS) Principle......Page 35
2.3.2 The LS Estimator......Page 36
2.3.4 Consistency and Asymptotic Normality of the LS Estimator......Page 37
2.3.5 Linear Semiparametric Model Inference......Page 38
2.3.6 Inferential Asymptotics......Page 39
2.3.7 Hypothesis Testing: Linear Equality Restrictions on β......Page 40
2.4 General Linear Model with Stochastic X......Page 42
2.4.1 Linear Model Assumptions......Page 43
2.4.3 LS Estimator Properties: Asymptotics......Page 44
2.4.4 ML Estimation of  and  under Conditional Normality......Page 45
2.4.5b Parametric Case......Page 46
2.5 Extremum (E) Estimation and Inference......Page 48
2.5.1 ML and LS Estimators Expressed in E Estimator Form......Page 49
2.5.2 Asymptotic Properties of E Estimators......Page 50
2.5.3 Inference Based on E Estimation......Page 51
2.5.4 Summary and Forward: E Estimators......Page 52
2.6 Selected References......Page 53
3.1 Introduction......Page 54
3.1.1 A Just-Determined Moment System with Random Sampling of Scalars......Page 55
3.2 Just-Determined Moment Systems, Random Sampling, and Method of Moments (MOM)......Page 57
3.2.1 General Asymptotic Properties......Page 58
3.2.2 Linear Model Semiparametric Estimation through Moment Equations......Page 59
3.2.3 MOM Conclusions......Page 60
3.3.1 GMM Framework......Page 61
3.3.2 GMM Linear Model Estimation......Page 62
3.3.2a Optimal GMM Weight Matrix......Page 63
3.3.2b Sampling Properties of Estimated Optimal GMM (EOGMM) Estimator......Page 64
3.3.2c Hypothesis Testing and Confidence Regions......Page 65
3.3.2d Additional Properties of the GMM Approach......Page 66
3.3.2e Summary and Forward: The GMM Approach......Page 67
3.4 Estimating Equations......Page 68
3.4.1 Duality between Estimating Equations (EEs) and E Estimators......Page 69
3.4.2 Linear Estimating Functions (EFs)......Page 70
3.4.3a Unbiasedness......Page 72
3.4.3b Optimal Estimating Functions (OptEFs): The Scalar Case......Page 74
3.4.3c OptEFs: The Multivariate Case......Page 75
3.4.4a Wald (W) and Z Tests and Confidence Regions......Page 77
3.4.4b Generalized Score (Lagrange Multiplier-Type) Tests and Confidence Regions......Page 78
3.4.4.c Pseudo-Likelihood Ratio Tests and Confidence Regions......Page 79
3.5 E Estimation with Instrumental Variables......Page 80
3.6 Summary and Forward......Page 81
3.7 Selected References......Page 82
PART II FORMULATION AND SOLUTION OF STOCHASTIC INVERSE PROBLEMS......Page 85
4.1  Introduction......Page 87
4.2 A Stochastic Linear Inverse Problem......Page 89
4.2.1 Addressing the Indeterminacy of Unknowns......Page 91
4.3.1 Nonparametric ML......Page 92
4.3.2 Empirical Likelihood (EL) Function for θ......Page 94
4.3.3 Comparing the Use of Estimating Functions in EE and EL Contexts......Page 96
4.3.4 The Functional Form of the EL Function......Page 98
4.3.5 Summary of the EL Concept......Page 99
4.3.6 Maximum Empirical Likelihood (MEL) Estimation of a Population Mean......Page 100
4.3.7 MEL Linear Model Estimation for Stochastic X......Page 103
4.4 Epilogue......Page 104
APPENDIX 4. A NUMERICAL EXAMPLE: COMPUTING MEL ESTIMATES......Page 105
5.2 MEL Inference: iid Case......Page 108
5.2.1 MEL Efficiency Property......Page 109
5.3 Empirical Example of MEL Estimation Based on Two Moments......Page 112
5.4.1 Empirical Likelihood Ratio Tests and Confidence Regions for c(θ)......Page 113
5.4.2 Wald Tests and Confidence Regions for c(θ)......Page 114
5.4.4 Z-Test of Inequality Hypotheses for the Value of c(θ)......Page 115
5.4.5 Testing the Validity of Moment Equations......Page 116
5.4.6 MEL Testing and Confidence Intervals for Population Mean......Page 117
5.4.7 Illustrative MEL Confidence Interval Example......Page 118
5.5 Concluding Comments......Page 119
5.6 Selected References......Page 121
6.1.1 Solutions to Systems of Estimating Equations and Kullback-Leibler Information......Page 122
6.2 Kullback-Leibler Information Criterion (KLIC)......Page 124
6.2.1 Relationship between Maximum Empirical Exponential Likelihood (MEEL) and KL Information......Page 126
6.2.1a Objective of MEEL......Page 127
6.3.1 The MEEL Estimator and Alternative Empirical Likelihood......Page 129
6.3.2 MEEL Asymptotics......Page 130
6.3.3a Testing H0 : c(θ)=r......Page 132
6.3.3c Testing the Validity of Moment Equations......Page 133
6.3.4 Contrasting the Use of Estimating Functions in EE and MEEL Contexts......Page 134
6.4.1 Combating Model Uncertainty: General Combining Formulations......Page 135
6.4.2 Example: A Combined Estimator......Page 137
6.4.2a Finite Sample Performance......Page 138
6.5 An Informative Reference Distribution......Page 139
6.6 Concluding Remarks......Page 141
6.7 Reader Idea Checklist......Page 142
6.8 Selected References......Page 143
Appendix 6.ARelationship between the Maximum Empirical Likelihood (MEL)\nObjective and KL Information......Page 144
APPENDIX 6.B NUMERICAL ILLUSTRATION OF MEEL AND MEL ESTIMATION OF A PROBABILITY DISTRIBUTION......Page 146
APPENDIX 6.C SHANNON’S ENTROPY – SOME HISTORICAL PERSPECTIVE......Page 148
PART III A FAMILY OF MINIMUM DISCREPANCY ESTIMATORS......Page 151
7.1 Introduction......Page 153
7.1.1 Family of Likelihood Functions......Page 154
7.2 The Cressie-Read (CR) Power Divergence Family......Page 155
7.3 Three Main Variants of I(p, q, γ)......Page 157
7.4 Minimum Power Divergence and Empirical Maximum Likelihood (EML) Estimation......Page 158
7.5.1 Test Statistics......Page 160
7.5.1a Moment Validity Tests......Page 161
7.5.1b Tests of Parameter Restrictions......Page 162
7.6 Concluding Remarks......Page 163
7.7 Selected References......Page 164
APPENDIX 7.A PROPOSITIONS, PROOFS, AND DEFINITIONS......Page 166
APPENDIX 7.B ENTROPY FAMILIES......Page 170
8.1 Introduction......Page 171
8.2 Design of Sampling Experiments......Page 172
8.3.1 Estimator MSE Performance......Page 174
8.3.2 Bias and Variance......Page 175
8.3.4 Size of Moment Validity Tests......Page 177
8.3.5 Confidence Interval Coverage and Expected Length......Page 178
8.4 Summary Comments......Page 179
8.5 Selected References......Page 181
APPENDIX 8.A COMPUTATIONAL ISSUES AND NUMERICAL APPROACH......Page 182
PART IV BINARY–DISCRETE CHOICE MINIMUM POWER DIVERGENCE (MPD) MEASURES......Page 185
9.1 Introduction......Page 187
9.2.1 Parametric......Page 189
9.2.2 Nonparametric Stochastic Inverse Problem Representation......Page 190
9.3 Minimum Power Divergence Class of CDFs for the Binary Response Model......Page 191
9.3.1 Applying MPD to Conditional Bernoulli Probabilities......Page 192
9.3.2 Conditional Reference Probabilities......Page 193
9.3.3 The Class of CDFs Underlying p......Page 195
9.3.4 Properties of the MPD Class of Probability Distribution Functions......Page 197
9.3.4a Moments......Page 199
9.4 Summary and Extensions......Page 200
9.5 Selected References......Page 201
9.A.1 Alternative form of E(g (W))......Page 202
9.A.2 Concavity of ln(F (w; q, γ)) and ln(1 − F (w; q, γ))......Page 203
10.1 Introduction......Page 205
10.2.1 Interpreting  as λ an Estimator of β......Page 206
10.2.2 Estimating the Marginal Probability Effects of Changes in Response Variables......Page 208
10.3 Asymptotic Estimator Properties......Page 209
10.3.1 Asymptotic Inference......Page 210
10.4.1 EML Estimation......Page 211
10.5 Sampling Performance......Page 213
10.5.1 Sampling Design......Page 214
10.5.2 Sampling Results......Page 216
10.6 Summary and Extensions......Page 218
10.7 Selected References......Page 219
APPENDIX 10.A ASYMPTOTIC PROPERTIES OF MPD(q, γ) – APPLICABILITY OF ASSUMPTIONS......Page 220
APPENDIX 10.B CONSISTENCY......Page 221
APPENDIX 10.C ASYMPTOTIC NORMALITY......Page 222
PART VOPTIMAL CONVEX DIVERGENCE......Page 223
11.1 Introduction......Page 225
11.2 Econometric Model and the Cressie-Read (CR) Family......Page 226
11.3 Choosing a Minimum Loss Estimation Rule......Page 227
11.3.1 Distance–Divergence Measures......Page 228
11.3.2 A Minimum Quadratic Risk Estimation Rule......Page 229
11.3.3 The Case of Two CR Alternatives......Page 230
11.3.4 Empirical Calculation of α......Page 231
11.4 Finite Sample Implication......Page 232
11.5 Estimator Choice, γ = (1, 0,−1)......Page 234
11.6 Sampling Performance......Page 235
11.7 Concluding Remarks......Page 236
11.8 Selected References......Page 237
APPENDIX 11.A A γ = (0,−1) SPECIAL CASE CONVEX ESTIMATION RULE......Page 238
TWELVE Epilogue......Page 239
CHAPTER 3......Page 241
CHAPTER 7......Page 242
CHAPTER 11......Page 243
Index......Page 245